The complex rational expression provided is:

$\frac{12x + 1z - 7}{111z + 2x} \div \frac{11z^2 - 77z}{1}$

Let's break this down and simplify it step by step.

Step 1: Rewrite the expression as a division problem

The expression can be rewritten as a multiplication problem by taking the reciprocal of the second fraction:

$\frac{12x + 1z - 7}{111z + 2x} \times \frac{1}{11z^2 - 77z}$

Step 2: Factor out common terms

1. In the second fraction $11z^2 - 77z$, you can factor out the common term $11z$: $11z^2 - 77z = 11z(z - 7)$

So the expression becomes:

$\frac{12x + z - 7}{111z + 2x} \times \frac{1}{11z(z - 7)}$

Step 3: Combine into a single fraction

Now multiply the two fractions:

$\frac{(12x + z - 7) \times 1}{(111z + 2x) \times 11z(z - 7)}$

This simplifies to:

$\frac{12x + z - 7}{11z(111z + 2x)(z - 7)}$

Final Simplified Expression:

The simplified form of the complex rational expression is:

$\frac{12x + z - 7}{11z(111z + 2x)(z - 7)}$

Let me know if you'd like more details on any of the steps!

Related Questions:

1. How can we simplify complex fractions more effectively?
2. How does factoring help in simplifying rational expressions?
3. Can complex rational expressions have undefined points?
4. What is the significance of common terms in simplifications?
5. What are the applications of simplifying rational expressions in real life?

Tip: When simplifying, always look for common factors to reduce the complexity early on!